Start saving your money to buy some dumbbells or a barbell, because come on, they work better. Used equipment is cheap if cost is an issue.

Adding muscle without barbells

If you absolutely refuse to lift a barbell (or even a dumbbell or kettlebell), then you could see some results from three to six months of upper-body work. That means a whole lot of chin-ups, pull-ups, dips, and push-ups (including divebomber or handstand variations). In order to avoid totally neglecting your lower body, do some squats, pistol squats, and back extensions as well.

I saw a modest improvement in my shoulders, upper arms, and upper back from four months of doing 3 maximal sets each of pull-ups and dips. I only added five or ten pounds of muscle, however. Three workouts a week with a minimum of 50 pull-ups and dips would be a fine start to a muscular upper body.

Eat

To get bigger, you must give your body the raw materials with which to make more of you. Eat a lot of vegetables and pastured meat, milk, and eggs. Focus on food quality--avoid soda, junk and deep-fried food, get grass-fed beef and dairy--but don't forget to simply eat a lot.

BTW - I don't want overstep my boundaries but to my knowledge a boy at 15 if he is that tall and its not common in the family he should probably be taken to the doctor to verify that he is not suffering from gigantism.